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In this paper we introduce the class of differentiable weights ω in the unit disc 𝔻 such that

where L is a positive constant.

The main result in this paper asserts that if ω is one of these weights, then the equivalence

holds for all 0 < p < ∞, 0 < q ≤ ∞and f an analytic function in 𝔻. Our results improve others due to Aleman, Siskakis, and Stević.

We also prove two results on harmonic conjugate functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)